Acta Materialia 53 (2005) 4237–4244

www.actamat-journals.com

Hydraulically pumped cone fracture in brittle solids

Herzl Chai a, Brian R. Lawn b,*

a Department of Solid Mechanics, Materials and Systems, Tel Aviv University, Tel Aviv, Israelb Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Building 301, Shipping and Receiving,

Gaithersburg, MD 20899-8500, USA

Received 27 April 2005; received in revised form 16 May 2005; accepted 16 May 2005Available online 11 July 2005

Abstract

An analysis of inner cone cracks in brittle solids subject to cyclic indentation in liquid is presented. These inner cone cracks, sonamed because they form well within the maximum contact circle, are postulated to be driven by a hydraulic pumping mechanism.Unlike their more traditional outer cone crack counterparts, inner cones do not appear in monotonic loading or even cyclic loadingin the absence of liquid. According to the hydraulic pumping postulate, an expanding contact engulfs surface fissures or flaws, clos-ing the crack mouths and squeezing entrapped liquid toward the subsurface tips, thereby enhancing downward penetration. Finiteelement modeling is used to analyze the stress and displacement fields in the vicinity of the growing cracks and to compute stress-intensity factors, using soda-lime glass loaded by a spherical indenter in water as a case study. A stepwise incrementing proceduredetermines the inner cone crack evolution, i.e., crack depth c as a function of number of indentation cycles n, for this system. Thepredicted c(n) response reproduces essential features of experimentally observed behavior, relative to companion outer cone cracks:notably, a sluggish start in the initial regions, where Hertzian stresses govern, followed by rapid acceleration as hydraulic pumpingactivates, ultimately dominating in the steady-state far-field.Published by Elsevier Ltd on behalf of Acta Materialia Inc.

Keywords: Contact fracture; Curved surfaces; Cyclic loading; Hydraulic pumping; Inner cone cracks

1. Introduction

The incidence of fracture from contact loading be-tween a thick brittle solid and a curved indenter is along-standing classical problem, beginning with Hertzin the 1880s [1] and intensifying over the past four dec-ades with the advancement of indentation fracturemechanics [2,3]. Most familiar are the shallow circum-ferential or ‘‘outer’’ cone or ring fractures that form out-side an expanding sphere contact in single-cycle loading:in brittle monoliths the location of these outer surfacerings typically lies within 15% of the contact circle [4];in brittle coating layers on compliant substrates the sur-face rings migrate much further out from the contact, as

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* Corresponding author. Tel.: +1 301 975 5775.E-mail address: brian.lawn@nist.gov (B.R. Lawn).

a result of superimposed flexural stresses [5]. More re-cently, with extension of experimentation to cyclic load-ing in liquid environments, a second kind of cone crackhas been observed to form well within the maximumcontact circle [6]. These so-called ‘‘inner’’ cone cracksare noticeably steeper and can propagate much deeperafter extended cycling, sometimes leading to penetrationfailure in thinner plates [7]. Examples from an earlierstudy are shown in Fig. 1 for cyclic loading with a tung-sten carbide (WC) spherical indenter on glass and porce-lain in water [6]. Inner cones form only in cyclic loadingin water, so they cannot be explained by an exclusivelytime-dependent mechanism such as chemically enhancedslow crack growth in the same way as outer cone cracks[6,8,9]. It would appear that some additional, cumula-tive mechanical driving force associated with the peri-odic ingress and egress of liquid must operate [7],

c.

Fig. 1. Cone cracks in brittle material subject to cyclic contact in waterwith WC spheres of radius r = 3.18 mm. Half-surface and sectionmicrographs of cracks in (a) glass (load P = 140 N, n = 104 cycles) and(b) porcelain (P = 500 N, n = 5 · 104). Shallow outer and steep innercracks are evident. Fretting zone due to slip at indenter/specimeninterface within contact can provide starting flaws for inner cone crackinitiation. Micrographs from Kim et al. [6].

4238 H. Chai, B.R. Lawn / Acta Materialia 53 (2005) 4237–4244

somewhat analogous to the pressure-induced infusion oflubricant into pitting cracks in metal surfaces duringrolling contact fatigue [10]. Inner cones can become adominant mode of failure in hard-coating bilayer sys-tems in repetitive contact loading in aqueous solutions,e.g., dental crowns in occlusion function, and are there-fore of great interest in biomechanical systems [7].

In this study, we analyze the mechanics of inner conecrack evolution in monolithic brittle solids from cycliccontact loading in liquid. Any such analysis must ex-plain how the inner cones initiate at the specimen sur-face well within the maximum contact circle and

penetrate a highly compressive stress field immediatelybeneath the expanding contact before reaching theindentation far-field. It must also account for a mechan-ical driving force on these cracks from intrusive liquid,over and above the usual slow crack growth effects, inorder to explain why these cracks are not apparent incyclic loading in air. The mechanism envisaged is onein which liquid is squeezed into surface fissures towardthe subsurface crack tips as the contact engulfs the flaw[11]. This is a complex fracture mechanics probleminvolving interaction of Hertzian contact and fluid pres-sure fields, and so we use finite element analysis to ob-tain numerical solutions for inner cone crack evolutionin a specific material under specific cyclic loading condi-tions – silicate glass cycled with a WC sphere of pre-scribed radius in water – for which experimental dataare available. We will show that a hydraulic pumpingmodel can account for basic data trends. We will alsoshow that once the cone cracks penetrate the near-con-tact zone and enter a tensile far-field the hydraulic driv-ing force peters out, and the crack growth characteristicsasymptotically approach a familiar analytical solutionfor center-loaded penny cracks driven solely by slowcrack growth.

2. Liquid entrapment mechanism

The contact variables of interest are depicted in theschematic of Fig. 2: a spherical indenter of radius r atload P and contact radius a, with maximum values Pm

and am. The contact is assumed to take place in a liquidenvironment over n cycles. Cone cracks located a dis-tance R from the contact axis form at an angle a tothe top surface and propagate to a length C or depthc = C sina. Tensile stresses occur outside a drop-shapedcompression zone of approximate radius a beneath theindenter, shown as the shaded area in Fig. 2. Outer conecracks (O) form at RO � 1.15am with aO � 22� [4], andhence are never subject to compression through the cyc-lic evolution. Inner cones (I) form at RI � 0.5am withai � 50� (Fig. 1), although some variations in these coor-dinates occur from material to material. This locationrepresents a compromise between counteracting factors:if RI were to be smaller than 0.5am then the crack wouldopen at lower loads but there would not be a large vol-ume of entrapped liquid to drive the fracture; if RI wereto be larger than 0.5am then the volume of entrappedfluid would be larger but the range of contact radiusover which this fluid could be forced into the crack be-fore reaching its maximum at am would be reduced.The cone angles closely approximate maximum princi-pal stress trajectories in the Hertzian field at maximumcontact, implying essentially tensile fracture (mode I)[12]. The inner cones experience small surface tensilestresses prior to engulfment (a < RI, Fig. 3(a)), but much

Liquid

Liquid

a

b

Fig. 3. Schematic demonstrating liquid entrapment model. Shadedarea beneath contact designates approximate compression zone.(a) Liquid enter crack prior to contact engulfment. (b) Liquid squeezestoward crack tip as contact expands, causing incremental extension.Cyclic contact repeats process, forcing more liquid into crack insuccessive cycles.

P,n

r

αc

a

C

SO

I

Liquid R

Fig. 2. Schematic of cone crack system in brittle solid. Sphericalindenter, sphere radius r, contact radius a, load P and cycles n. Conecracks of length c = Csina, angle a to surface, located distance R fromcontact axis. Label O designates outer cones, I inner cones. Shadedarea indicates compression zone below contact. Contact occurs inliquid. (Linear crack profiles are simplistic representations of actualpaths in Fig. 4.)

H. Chai, B.R. Lawn / Acta Materialia 53 (2005) 4237–4244 4239

higher surface compressive stresses after engulfment(a > RI, Fig. 3(b)). However, even inner cones will expe-rience tensile stresses at their extremities after engulf-ment if they can somehow grow sufficiently large so asto penetrate the compression zone.

We have mentioned that inner cone cracks developfully only in cyclic loading in liquid, so that there mustbe some superposed mechanical force associated with li-quid intrusion. Consider a starting flaw located at dis-tance RI from the contact axis on the surface. As theload increases, the contact at first opens the crack,allowing liquid to enter by capillary action (Fig. 3(a)).When the contact circle becomes coincident with theflaw (a = RI), the crack mouth is sealed (although theremay exist local fluid reservoirs at the indenter/specimeninterface associated with any surface roughness). Fur-ther increase in load engulfs the crack (a P RI), andthe crack becomes subject to mouth closure stresseswithin the drop-shaped compression zone (Fig. 3(b)).Ingress or egress of liquid is then restricted until the in-denter is unloaded. Mode I crack opening is induced byfluid pressure by one or both of two possible mecha-nisms [11]: (i) liquid is forced into surface fissures, pro-viding a wedging force in proportion to the Hertzianpressure; (ii) liquid is forced downward after engulfmentin a ‘‘pinching’’ action, closing the crack mouth butopening the tip. The second mode is more consistentwith experimental observations where the mouths ofwell-developed inner cones are observed to close andthe far ends open up during loading half-cycles [7].

3. Computation of stress-intensity factors

3.1. FEA model

In the case of outer cone cracks, analytical solutionsfor incremental cyclic crack growth are available in boththe small-flaw initiation and long-crack propagationstages, based on slow crack growth models [6,13]. Forinner cones, there currently exist no such analytical solu-tions, owing to the superposition of complex mechanicaldriving forces associated with ingressing fluid in the sur-face cracks.

Accordingly, finite element analysis (FEA) tech-niques are here used to demonstrate the feasibility ofthe hydraulic pumping mechanism. The basic FEA

OO

I I

Fig. 4. Calculated crack paths for outer and inner cone cracks in soda-lime glass indented with WC sphere. Note curvature of crack pathsnear surface, relative steepness of inner cone cracks.

100

1

0.1

0.3

0.6

3

101 103

Stre

ss in

tens

ity f

acto

r, K

(M

Pa. m

1/2 )

Crack depth, c (µm)

Glass/water Pm = 120 N

r = 1.58 mm

K0

KIKI

KI+

KI–

Fig. 5. Mode I stress-intensity factors K as function of crack depth c

for glass indented in water with WC sphere. KO is SIF for outer conecrack. KI is SIF for inner cone crack without fluid pressure, includingcontribution from tensile field after engulfment. Dashed curve isequivalent K�

I with no engulfment. KþI is SIF with fluid pressure

included. Dashed lines at right are asymptotic K � c�3/2 limits for far-field cracks.

4240 H. Chai, B.R. Lawn / Acta Materialia 53 (2005) 4237–4244

procedure has been outlined in previous studies [5,14]. Aflat glass half-space (Young�s modulus E = 70 GPa andPoisson�s ratio m = 0.22) is indented with a WC sphere(E = 614 GPa and m = 0.22) of radius r = 1.58 mm infrictionless axisymmetric contact. In accord with exist-ing experimental data [7], the indenter is loaded in smallsteps to produce an expanding contact a relative to thesurface crack locations RI and RO, up to a maximumcontact am = 130 lm at corresponding load Pm =120 N. Calculations are performed for outer and innercone cracks separately, ignoring any potential interac-tion between the two. The surface cracks are allowedto extend downward and outward in increments Dc,beginning with short flaws of depth c = 1 lm normalto the surface and with Dc � c at all stages of growth.The mesh is reconfigured at each crack increment, withthe grid typically consisting of a minimum 80 equispacedelements along the crack walls (a sufficient number toensure convergence in the FEA solutions). Stress-inten-sity factors at each crack depth c are calculated from no-dal displacements at the crack walls using the Irwinparabolic crack-tip COD relation [14]. The kink anglea of each crack increment is determined by an angularsearch algorithm as that which optimizes mode I exten-sion (i.e., KII = 0) over the entire load spectrum (i.e.,P 6 Pm). In general, it is found that the cracks followa somewhat curved path in the small c region, wherestress gradients are high, and subsequently straightenout into nearly straight-sided (truncated) cones in thelarge c region as the cracks approach the far-field. Ofcourse, the angular kink search routine becomes lessonerous once the crack straightens out.

These calculations are performed for cracks withoutand with entrapped fluid, in the latter case by applyingsuperposed uniform normal forces at the nodes alongthe crack walls to simulate fluid pressure under constantcrack volume conditions (fluid incompressibilityapproximation), as described below.

3.2. Outer cone cracks

First we examine the outer cones. FEA computationsfor a similar crack system in punch loading have beenwell described in the literature by Kocer and Collins[15], and we present them here only to confirm the valid-ity of the calculations. Following experimental observa-tion [7], we set RO = 1.15am = 150 lm at Pm = 120 N.As indicated, the outer cone crack always remains out-side the maximum contact circle, with maximum open-ing at a = am. In this case, confining fluid pressure isnever an issue.

The calculation begins with a normal starting flaw ofsize cO = 1 lm, commensurate with the scale of smallflaws in glass surfaces. The flaw is allowed to incrementits depth through a specified amount DcO. The algorithmsearches for the kink angle aO that maintains the crack

in pure mode I. For outer cones, the maximum KO is al-ways achieved at maximum load P = Pm, where the ten-sile stress field is greatest. Crack increments arecontinued sequentially. The crack first forms a shallowcollar before curving outward into its characteristictruncated cone geometry. For soda-lime glass, the coneangle in the far-field is aO = 22�. The calculated crackpath is indicated by the outer trajectories in Fig. 4.

The function KO(cO) for outer cones determined fromsuch calculations is plotted as the uppermost curve inFig. 5. Note that this curve passes through a maximumat cO � 8 lm or cO/am � 0.06. The broad shape of thiscurve and position of the maximum are similar to thecalculations of Kocer and Collins [15]. Note also that

0.1

00 20 40 60 80

0.2

0.3

Crack coordinate, S (µm)

Cra

ck o

peni

ng d

ispl

acem

ent,

u (µ

m)

P = PI

P = P*

cI = 50 µm

Fig. 6. Crack opening displacements u as function of distance S alongcrack path for inner cone cracks in glass, without liquid at P = PI andwith liquid pressure at P = P*, volume V = V0, for crack depthcI = 50 lm. Note how mouth closure in second case opens tip region,enhancing field.

H. Chai, B.R. Lawn / Acta Materialia 53 (2005) 4237–4244 4241

the long-crack tail of this curve tends asymptotically tothe familiar relation KO / P=c3=2O (light dashed line) forwell-developed cones [2,3].

3.3. Inner cone cracks

3.3.1. Cracks without fluid

For inner cones, we set RI = 0.5am = 65 lm forPm = 120 N as per experimental observation [7]. We firstperform this calculation in absence of any fluid pressureat the crack walls. Following the procedure in Section3.2, start with an initial surface flaw at cI = 1 lm and al-low stepwise incremental extensions DcI. The K termsare computed at each step from the crack-wall nodal dis-placements behind the tip. The behavior is now morecomplex, because the configuration of maximum KI doesnot necessarily always occur at the engulfment load PI

(Fig. 3(a)), nor at the maximum load Pm (Fig. 3(b)).In other words, inner cone cracks may in some casescontinue to extend after first engulfment. To handle thiscomplication, we first retain the angle aI from the pre-ceding crack increment and determine the load P within0 6 P 6 Pm at which KII = 0. Then we repeat the com-putation for various aI and choose the value of P = P*that maximizes KI. For the case study here, this config-uration occurs at P* < Pm for cI < 1.5am and at P* = Pm

for crack depths cI > 1.5am approximately. The calcu-lated crack paths are indicated by the inner trajectoriesin Fig. 4. As with its outer counterpart, the inner conecrack curves away from normal near the surface, butthen straightens out to aI � 50� over the long-cracklength for this particular flaw location. The fact that thiscrack angle is quite different to that of the outer cone be-yond the collar region is a manifestation of the fact thatthe driving force on the crack can indeed continue to in-crease during the engulfment stage.

The function KI(cI) for inner cones without fluid pres-sure is included in Fig. 5. The dashed curve K�

I indicatesthe same function but without engulfment, i.e., stoppingthe loading at P = PI (Fig. 3(a)). This latter curve hasthe same basic shape as KO but with a downward andleftward shift. The hump in the KI function at cI � 20 l-m = 0.15am is then due exclusively to additional crackgrowth beyond engulfment. Again, this is because longercracks begin to experience some increase in tensile stressintensity in the tip regions as they begin to penetrate theimmediate compressive zone. Such additional post-engulfment augmentation, although substantial, is stillnot large enough to make the inner cone cracks compet-itive with their outer counterparts in the short-crackregion.

3.3.2. Cracks with fluid pressure

Now consider the effect of superposing a fluid pres-sure on the inner cone crack walls during loading. Thesequence of computation is as follows. For any given

crack depth cI and kink angle aI evaluated in the previ-ous section, determine the crack opening displacementsu as a function of coordinate S along the crack path(Fig. 2) immediately prior to first engulfment atP = PI (a = RI). Calculate the initial crack volume V0

from the area under the u(S) curve (allowing for theever-increasing base radius with increasing S). Then in-crease the load to P = P* (a = a*), while maintainingfluid volume constant at V = V0 (incompressible liquid).This latter is achieved by introducing equal normalforces over equispaced mesh nodes along the crackwalls, equivalent to an internal hydrostatic pressure,starting from zero and incrementing (or decrementing)the forces until the requisite crack volume V0 is attained.Finally, calculate the stress-intensity factor Kþ

I withsuperposed fluid pressure from the new nodaldisplacements.

An example of the change in crack profile for a crackof length C = 70 lm (depth cI = 50 lm) and V = V0 isshown in Fig. 6. Note how the crack closes near themouth in the engulfed crack at P = P*, simultaneouslyopening up the tip region and thereby enhancing KI.Note also the parabolic contours close to the cracktip, in accordance with the Irwin crack tip near-field.

The calculations are then repeated for several crackdepths cI, determining a new initial fluid volume V0 ateach cI. This yields the function Kþ

I ðcIÞ for inner coneswith fluid pressure included in Fig. 5. The contributionfrom hydraulic pressure is thus the differentialDKI ¼ Kþ

I � KI (not plotted in Fig. 5). An importantpart of the hydraulic effect is the systematic increase influid volume V0 with each successive cycle, as liquid iscontinually forced in and out of the crack mouth duringunloading and reloading. Hence this contribution is nota factor in monotonic loading (i.e., DKI = 0). Note that

20

10

40

30

60

50

70

00 20 40 60 80 100 120

Crack depth, c (µm)

Liq

uid

pres

sure

in c

rack

(M

Pa)

A

B

C

Fig. 7. Fluid pressure in crack. Solid curve is pressure needed tomaintain V = V0 at load P = P* for a crack of fixed depth c. Dashedcurves indicate how pressure would diminish if cracks at A, B and Cwere to be allowed to extend at constant fluid volume and fixedP = P*.

4242 H. Chai, B.R. Lawn / Acta Materialia 53 (2005) 4237–4244

the quantity DKI again does not come into play untilcI > 17 lm = 0.13am. (Recall, however, that we areignoring possible wedging forces from enforced liquidpenetration at the Hertzian pressure in this short-crackregion.) The hydraulic term DKI considerably enhancesthe stress-intensity factor in this intermediate crack sizeregion. Such enhancement is evident in Fig. 7 as the plotof the equilibrium fluid pressure needed to maintainV = V0 at load P = P* as a function of fixed crack depthc, shown as the solid curve. This bell-shaped curve islimited to the region 17 lm < c < 110 lm, with a maxi-mum at c = 35 lm. In principle, once the crack becomessufficiently large that its tip penetrates beyondc > 110 lm in Fig. 7, corresponding to the region be-yond the compression zone at max load Pm (Fig.3(b)), the fluid pressure can become negative and there-by act to restrain the crack. In reality, the water may beexpected to cavitate at this point, effectively wiping outthe hydraulic effect. At this stage the inner cone crackis well developed and subject entirely to the Hertzianfar-field, with familiar asymptotic KI / P=c3=2I depen-dence (dashed line in Fig. 5).

4. Crack extension conditions

Consider now how much crack growth takes place inany given contact cycle. At this stage, it is necessary tointroduce a crack extension condition. For growth un-der equilibrium conditions, extension occurs at K = Kc,where Kc is the toughness. However, Kþ

I ðcIÞ in Fig. 5does not exceed the toughness Kc = 0.75 MPa m1/2 forsoda-lime glass until c � 30 lm, so small flaws couldnever extend under such conditions. In reality, crackgrowth in brittle materials in liquid environments occurs

at K < Kc under kinetic conditions, down to K levels be-low Kc/3. This kinetic condition is reasonably well rep-resented by a power law crack velocity relation [16,17]

t ¼ t0ðK=KcÞN ð1Þ

with N = 17.9 an exponent and t0 = 2.4 mm s�1 a coef-ficient for soda-lime glass in water [18]. Under these con-ditions, inner cone cracks represented in Fig. 5 aresubject to significant extension.

The velocity equation can then be used to calculatethe evolution for outer cone cracks over a number ofcontact cycles n at frequency f via a seemingly straight-forward numerical time-incrementing procedure.The number of cycles Dn to propagate a crack of givendepth cO through DcO = tDt sinaO can be writtenDn = fDt/b = fDcO/bt sinaO, where the K term used toevaluate t in Eq. (1) is calculated at P = P* (maximummode I configuration) and b is a coefficient (<1) to allowfor the fact that the crack is not subject to maximum K

over an entire cycle [6]. With evaluations of this kind atspecific crack lengths, Dn can be integrated numericallyto determine c(n).

For inner cone cracks, the procedure becomes morelaborious, because the term DKI from fluid pressure issuperposed onto the K-field. Recall that this contribu-tion is only important during the intermediate stage ofgrowth in Fig. 5, corresponding to penetration throughthe compression zone. The complication arises becausethe entrapped fluid within the inner cone crack walls willbe redistributed over a greater crack length as extensionproceeds, causing the pressure to relax during extensionand thereby diminishing DKI within any single loadingcycle. This relaxation is illustrated for three cracksdepths (A, B and C) as dashed lines in Fig. 7. To accom-modate this relaxation, we adopt the following proce-dure: (i) as above, for a given initial crack depth cI,determine the initial crack volume V0 at first engulfmentand calculate KI at P = P* while keeping V = V0; (ii) atthe selected value of cI, extend the crack depth in smallstepwise increments dcI at P and V constant, recalculat-ing KI at each point, as shown in Fig. 8 (how far KI dec-lines down these curves during any one complete cyclewill depend on the loading frequency); (iii) integratedcI/t in conjunction with Eq. (1) until the condition

Z DcI

0

dcI=t sin aO ¼ b=f

is satisfied, thereby identifying the extension DcI within acomplete cycle; (iv) repeat for selected values of cI andthereby determine the functional dependence DcI(cI);(v) numerically integrate DcI over all cycles to determinecI(n).

The resulting cI(n) functions for a specific case study,glass/water for maximum load Pm = 120 N and fre-quency f = 1 Hz, are plotted as the solid lines in

1

00 5 10 15

0.4

0.8

0.2

0.6

Stre

ss in

tens

ity f

acto

r, K

I (M

Pa. m

1/2 )

Incremental crack extension, δc (µm)

Glass/water

Pm = 120 N

r = 1.58 mmcI = 35 µm

cI = 22 µm

cI = 160 µm

cI = 65 µm

Fig. 8. Plot of KI(dcI) for selected crack depths cI for inner cone cracksin glass at fixed load P = P*.

Outer

Inner

100 101 102 103 104 105 106

Cra

ck d

epth

, c (

µm)

Number of cycles, n

Glass/water

Pm = 120 N

f = 1 Hz

r = 1.58 mm

10

30

60

300

600

100

1000

Fig. 9. Predictions of crack evolution functions c(n) for cyclicindentations with WC sphere on soda-lime glass in water, shown assolid curves. Data points are corresponding experimental data (threecracks each mode), filled symbols inner cone cracks and unfilledsymbols outer cone cracks [7]. Dashed curve represents growth of innercone crack in absence of fluid pressure.

H. Chai, B.R. Lawn / Acta Materialia 53 (2005) 4237–4244 4243

Fig. 9. This example is chosen because of availability ofcorresponding experimental data [7]. Data for outercone cracks are shown as the unfilled symbols, for innercone cracks as the filled symbols. The theoretical curvesare evaluated using an adjusted b = 0.1 to give a best fitto these data. The curves reproduce the main trends inthe experimental data: for outer cones, initiation withinthe first cycle, followed by steady growth thereafter intoa fully developed cone; for inner cones, relatively slowgrowth in the initial, short-crack region followed by anabrupt hydraulically driven spurt at n = 600 cycles inthe intermediate region, slowing finally into a steadygrowth phase in the long-crack region. This last regiontends asymptotically to the far-field limiting dependence

c � n2/3N (N = 17.9), obtained by combining the K �P/c3/2 relation for center-loaded penny cracks with thevelocity function t � KN [6]. The dashed curve in Fig.9 represents the growth of inner cone cracks withoutfluid pressure. Note whereas growth is still possible inthe absence of hydraulic effects, the predicted growthrate is too slow in relation to observed behavior.

5. Discussion

We have presented a model here, foreshadowed byothers [11], of hydraulically assisted crack growth inbrittle solids in cyclic contact with spherical indentersin liquids. The cracks of interest are penetrative innercones within the maximum contact circle, ordinarilynot observed in monotonic loading in any environmentor in cyclic loading in the absence of liquid. Themechanics are complicated by the fact that the innercone cracks may continue to grow even after engulfmentby the expanding contact, where compressive stresses inthe immediate subsurface contact zone act to close thecrack mouth yet tensile stresses just beyond this zoneopen the crack mouth. After an initial period of slowgrowth from Hertzian stresses alone, liquid entrapmentdelivers a fluid pressure which drives the crack at anaccelerated rate. Continued reloading enables more li-quid to enter in successive cycles to build up the pressurewith each cycle, counterbalancing the near-contact clo-sure forces. The model envisions progressive squeezingof entrapped liquid towards the subsurface crack tip ina kind of pumping mode, driving inner ring cracksthrough an otherwise inhibiting compressive contactzone into full cones. Once they penetrate into the far-field the inner cone cracks are driven by remote tensilestresses. Because of the complexity of the cyclic prob-lem, involving periodic crack extension from superposedmechanical and chemical forces, resort to FEA has herebeen made to evaluate the inner cone evolution. Analy-sis of a specific case study, that of contact of a glass sur-face with a WC sphere in water, reproduces the mainfeatures of the inner cone evolution in Fig. 9: namely,a slow start relative to its outer cone crack counterpart,but subsequent acceleration with cycling to become thedominant mode of fracture. Ultimately, in the large-crack region all cone cracks, inner and outer, tendasymptotically to a simple far-field limit c � n2/3N.

The principal value of the present FEA analysis is ademonstration of the feasibility of the hydraulic pump-ing model. At the same time, we acknowledge certainlimitations of the current analysis. We have conductedjust one case study, that of a specific material/inden-ter/environment system. The role of such variables assphere radius r, maximum load Pm and cyclic frequencyf would require a great deal more unwieldy, time-con-suming numerical analysis. For this reason, we would

4244 H. Chai, B.R. Lawn / Acta Materialia 53 (2005) 4237–4244

not advocate FEA as means for predicting generalbehavior. Quite apart from inaccuracies arising fromcrack increment problems, there are several assumptionsthat are open to question: (i) The hydraulic model isbased entirely on a squeezing mechanism whereby intru-sive liquid is forced toward the crack tip by closure stres-ses at the mouth. We have ignored possible wedgingstresses from forced entry of liquid from local reservoirsat a rough indenter/specimen interface into the crackmouth under Hertzian pressure. Such wedging actionis more likely to occur in the small flaw region, therebyenhancing and broadening the hump in the Kþ

I curve inFig. 5. (ii) Possible negative fluid pressures in the long-crack region, where contact engulfment actually actsto increase rather than decrease the confining volume,have been neglected. It has been simply assumed thatno such stresses will activate, because of cavitation ofthe entrapped liquid. In reality, cavitation will not occuruntil the negative pressure has achieved some non-zerovalue, thereby acting to inhibit the crack extension inthis region. (iii) A fixed location RI = 0.5am has been as-sumed for the inner surface ring. This midway locationbalances counteracting factors: higher RI/am would in-crease the stress intensity acting on the surface flawsprior to engulfment; lower RI/am would increase thestress intensity after engulfment. (iv) Interactions be-tween the inner cone and preceding outer cone havebeen ignored in the computation. Such interactionscould reduce the K-field on an adjacent inner cone crack,further contributing to a shift toward lower RI/am.(v) An attempt has been made to maintain small crackincrements, especially in the short-crack region. Stressinhomogeneities are extremely large in this region, soinaccuracies in the increment size and angle are likelyto be greatly magnified in the resultant K determinations[19]. Consideration of these factors as a whole suggeststhat our calculations, if anything, probably underesti-mate the scale of the hydraulic pumping effect.

The crack law used here to describe the rate depen-dence of crack growth is the power-law velocity functiont(K) in Eq. (1) [17]. Such a function is able to accountwholly for the fatigue responses of outer cone cracks [6].In principle, we may apply the same law to systems inany liquid environment, simply by changing the exponentand coefficient in this function. However, a simple-powerlaw relation does not hold for all liquids, e.g., alkanes,where the t(K) curve exhibits multiple branches, attribut-able to diffusion rates of trace water within the alkane liq-uids [20]. This simplymeans that the time integration over

the indentation cycling would have to be conducted en-tirely numerically. Somematerials like silicon are not sus-ceptible to slow crack growth at all. The inner cone crackscould extend only if the Kþ

I ðcÞ curve in Fig. 5 were to ex-ceed Kc. In principle, any fatigue would then be solelymechanical, from the hydraulic pumping effect alone,i.e., from continual cyclic augmentation of the entrappedfluid volume V0 within the crack walls. Such equilibriumgrowth could in principle be induced by increasing the ap-plied load to extremely high values,manifested inFig. 5 asa scaling up of the Kþ

I ðcÞ curve.The inner cones, by virtue of their steep angles to the

surface, can become a dominant mechanism of failure inthin specimens or in bilayers, e.g., dental crowns [7].Extension of the current analysis to that case is underway.

Acknowledgments

We acknowledge useful discussions with SheldonWiederhorn, Yu Zhang, Yeon-Gil Jung and SanjitBhowmick. This work was supported by a grant fromthe U.S. National Institute of Dental and CraniofacialResearch (PO1 DE10976).

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